Copied to
clipboard

G = C5×D27order 270 = 2·33·5

Direct product of C5 and D27

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C5×D27, C27⋊C10, C1352C2, C45.2S3, C15.2D9, C9.(C5×S3), C3.(C5×D9), SmallGroup(270,1)

Series: Derived Chief Lower central Upper central

C1C27 — C5×D27
C1C3C9C27C135 — C5×D27
C27 — C5×D27
C1C5

Generators and relations for C5×D27
 G = < a,b,c | a5=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >

27C2
9S3
27C10
3D9
9C5×S3
3C5×D9

Smallest permutation representation of C5×D27
On 135 points
Generators in S135
(1 112 82 55 48)(2 113 83 56 49)(3 114 84 57 50)(4 115 85 58 51)(5 116 86 59 52)(6 117 87 60 53)(7 118 88 61 54)(8 119 89 62 28)(9 120 90 63 29)(10 121 91 64 30)(11 122 92 65 31)(12 123 93 66 32)(13 124 94 67 33)(14 125 95 68 34)(15 126 96 69 35)(16 127 97 70 36)(17 128 98 71 37)(18 129 99 72 38)(19 130 100 73 39)(20 131 101 74 40)(21 132 102 75 41)(22 133 103 76 42)(23 134 104 77 43)(24 135 105 78 44)(25 109 106 79 45)(26 110 107 80 46)(27 111 108 81 47)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 81)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(83 108)(84 107)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)(109 115)(110 114)(111 113)(116 135)(117 134)(118 133)(119 132)(120 131)(121 130)(122 129)(123 128)(124 127)(125 126)

G:=sub<Sym(135)| (1,112,82,55,48)(2,113,83,56,49)(3,114,84,57,50)(4,115,85,58,51)(5,116,86,59,52)(6,117,87,60,53)(7,118,88,61,54)(8,119,89,62,28)(9,120,90,63,29)(10,121,91,64,30)(11,122,92,65,31)(12,123,93,66,32)(13,124,94,67,33)(14,125,95,68,34)(15,126,96,69,35)(16,127,97,70,36)(17,128,98,71,37)(18,129,99,72,38)(19,130,100,73,39)(20,131,101,74,40)(21,132,102,75,41)(22,133,103,76,42)(23,134,104,77,43)(24,135,105,78,44)(25,109,106,79,45)(26,110,107,80,46)(27,111,108,81,47), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(109,115)(110,114)(111,113)(116,135)(117,134)(118,133)(119,132)(120,131)(121,130)(122,129)(123,128)(124,127)(125,126)>;

G:=Group( (1,112,82,55,48)(2,113,83,56,49)(3,114,84,57,50)(4,115,85,58,51)(5,116,86,59,52)(6,117,87,60,53)(7,118,88,61,54)(8,119,89,62,28)(9,120,90,63,29)(10,121,91,64,30)(11,122,92,65,31)(12,123,93,66,32)(13,124,94,67,33)(14,125,95,68,34)(15,126,96,69,35)(16,127,97,70,36)(17,128,98,71,37)(18,129,99,72,38)(19,130,100,73,39)(20,131,101,74,40)(21,132,102,75,41)(22,133,103,76,42)(23,134,104,77,43)(24,135,105,78,44)(25,109,106,79,45)(26,110,107,80,46)(27,111,108,81,47), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(109,115)(110,114)(111,113)(116,135)(117,134)(118,133)(119,132)(120,131)(121,130)(122,129)(123,128)(124,127)(125,126) );

G=PermutationGroup([(1,112,82,55,48),(2,113,83,56,49),(3,114,84,57,50),(4,115,85,58,51),(5,116,86,59,52),(6,117,87,60,53),(7,118,88,61,54),(8,119,89,62,28),(9,120,90,63,29),(10,121,91,64,30),(11,122,92,65,31),(12,123,93,66,32),(13,124,94,67,33),(14,125,95,68,34),(15,126,96,69,35),(16,127,97,70,36),(17,128,98,71,37),(18,129,99,72,38),(19,130,100,73,39),(20,131,101,74,40),(21,132,102,75,41),(22,133,103,76,42),(23,134,104,77,43),(24,135,105,78,44),(25,109,106,79,45),(26,110,107,80,46),(27,111,108,81,47)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135)], [(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(28,41),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(56,81),(57,80),(58,79),(59,78),(60,77),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(83,108),(84,107),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101),(91,100),(92,99),(93,98),(94,97),(95,96),(109,115),(110,114),(111,113),(116,135),(117,134),(118,133),(119,132),(120,131),(121,130),(122,129),(123,128),(124,127),(125,126)])

75 conjugacy classes

class 1  2  3 5A5B5C5D9A9B9C10A10B10C10D15A15B15C15D27A···27I45A···45L135A···135AJ
order1235555999101010101515151527···2745···45135···135
size127211112222727272722222···22···22···2

75 irreducible representations

dim1111222222
type+++++
imageC1C2C5C10S3D9C5×S3D27C5×D9C5×D27
kernelC5×D27C135D27C27C45C15C9C5C3C1
# reps114413491236

Matrix representation of C5×D27 in GL2(𝔽271) generated by

1870
0187
,
231203
68163
,
183259
17188
G:=sub<GL(2,GF(271))| [187,0,0,187],[231,68,203,163],[183,171,259,88] >;

C5×D27 in GAP, Magma, Sage, TeX

C_5\times D_{27}
% in TeX

G:=Group("C5xD27");
// GroupNames label

G:=SmallGroup(270,1);
// by ID

G=gap.SmallGroup(270,1);
# by ID

G:=PCGroup([5,-2,-5,-3,-3,-3,752,237,3003,138,4504]);
// Polycyclic

G:=Group<a,b,c|a^5=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D27 in TeX

׿
×
𝔽